1. Reasoning Under Uncertainty
Artificial Intelligence
CMSC 25000
February 15, 2007

2. Agenda Motivation Probability and Bayes’ Rule
Reasoning with uncertainty
Medical Informatics
Probability and Bayes’ Rule
Bayesian Networks
Noisy-Or
Decision Trees and Rationality
Conclusions

3. Uncertainty Search and Planning Agents Real World:
Assume fully observable, deterministic, static
Real World:
Probabilities capture “Ignorance & Laziness”
Lack relevant facts, conditions
Failure to enumerate all conditions, exceptions
Partially observable, stochastic, extremely complex
Can't be sure of success, agent will maximize
Bayesian (subjective) probabilities relate to knowledge

4. Motivation Uncertainty in medical diagnosis
Diseases produce symptoms
In diagnosis, observed symptoms => disease ID
Uncertainties
Symptoms may not occur
Symptoms may not be reported
Diagnostic tests not perfect
False positive, false negative
How do we estimate confidence?

5. Motivation II Uncertainty in medical decision-making
Physicians, patients must decide on treatments
Treatments may not be successful
Treatments may have unpleasant side effects
Choosing treatments
Weigh risks of adverse outcomes
People are BAD at reasoning intuitively about probabilities
Provide systematic analysis

6. Probability Basics The sample space:
A set Ω ={ω1, ω2, ω3,… ωn}
E.g 6 possible rolls of die;
ωi is a sample point/atomic event
Probability space/model is a sample space with an assignment P(ω) for every ω in Ω s.t. 0<= P(ω)<=1; Σ ωP(ω) = 1
E.g. P(die roll < 4)=1/6+1/6+1/6=1/2

7. Random Variables
A random variable is a function from sample points to a range (e.g. reals, bools)
E.g. Odd(1) = true
P induces a probability distribution for any r.v X:
P(X=xi) = Σ{ω:X(ω)=xi}P(ω)
E.g. P(Odd=true)=1/6+1/6+1/6=1/2
Proposition is event (set of sample pts) s.t. proposition is true: e.g. event a= A(ω)=true

8. Why probabilities?
Definitions imply that logically related events have related probabilities
In AI applications, sample points are defined by set of random variables
Random vars: boolean, discrete, continuous

9. Prior Probabilities Prior probabilities: belief prior to evidence
E.g. P(cavity=t)=0.2; P(weather=sunny)=0.6
Distribution gives values for all assignments
Joint distribution on set of r.v.s gives probability on every atomic event of r.v.s
E.g. P(weather,cavity)=4x2 matrix of values
Every question about a domain can be answered with joint b/c every event is a sum of sample pts

10. Conditional Probabilities
Conditional (posterior) probabilities
E.g. P(cavity|toothache) = 0.8, given only that
P(cavity|toothache)=2 elt vector of 2 elt vectors
Can add new evidence, possibly irrelevant
P(a|b) = P(a^b)/P(b) where P(b) ≠0
Also, P(a^b)=P(a|b)P(b)=P(b|a)P(a)
Product rule generalizes to chaining

11. Inference By Enumeration

12. Inference by Enumeration

13. Inference by Enumeration

14. Independence

15. Conditional Independence

16. Conditional Independence II

17. Probabilities Model Uncertainty
The World - Features
Random variables
Feature values
States of the world
Assignments of values to variables
Exponential in # of variables
possible states

18. Probabilities of World States
: Joint probability of assignments
States are distinct and exhaustive
Typically care about SUBSET of assignments
aka “Circumstance”
Exponential in # of don’t cares

19. A Simpler World 2^n world states = Maximum entropy
Know nothing about the world
Many variables independent
P(strep,ebola) = P(strep)P(ebola)
Conditionally independent
Depend on same factors but not on each other
P(fever,cough|flu) = P(fever|flu)P(cough|flu)

20. Probabilistic Diagnosis
Question:
How likely is a patient to have a disease if they have the symptoms?
Probabilistic Model: Bayes’ Rule
P(D|S) = P(S|D)P(D)/P(S)
Where
P(S|D) : Probability of symptom given disease
P(D): Prior probability of having disease
P(S): Prior probability of having symptom

21. Diagnosis Consider Meningitis: Disease: Meningitis: m
Symptom: Stiff neck: s
P(s|m) = 0.5
P(m) =0.0001
P(s) = 0.1
How likely is it that someone with a stiff neck actually has meningitis?

22. Modeling (In)dependence
Simple, graphical notation for conditional independence; compact spec of joint
Bayesian network
Nodes = Variables
Directed acyclic graph: link ~ directly influences
Arcs = Child depends on parent(s)
No arcs = independent (0 incoming: only a priori)
Parents of X =
For each X need

23. Example I

24. Simple Bayesian Network
MCBN1
Need:
P(A)
P(B|A)
P(C|A)
P(D|B,C)
P(E|C)
Truth table 2
2*2
2*2*2
A = only a priori
B depends on A
C depends on A
D depends on B,C
E depends on C A B C D E

25. Simplifying with Noisy-OR
How many computations?
p = # parents; k = # values for variable
(k-1)k^p
Very expensive! 10 binary parents=2^10=1024
Reduce computation by simplifying model
Treat each parent as possible independent cause
Only 11 computations
10 causal probabilities + “leak” probability
“Some other cause”

26. Noisy-OR Example A B Pn(b|a) = 1-(1-ca)(1-L) Pn(b|a) = (1-ca)(1-L)
Pn(b|a) = 1-(1 -L) = L = 0.5
Pn(b|a) = (1-L)
P(B|A)
b b
Pn(b|a) = 1-(1-ca)(1-L)=0.6
(1-ca)(1-L)=0.4
(1-ca) =0.4/(1-L)
=0.4/0.5=0.8
ca = 0.2 a

27. Noisy-OR Example II Full model: P(c|ab)P(c|ab)P(c|ab)P(c|ab) & neg A B
Noisy-Or: ca, cb, L
Assume:
P(a)=0.1
P(b)=0.05
Pn(c|ab)=0.3
ca= 0.5
Pn(c|b) = 0.7 C
Pn(c|ab) = 1-(1-ca)(1-cb)(1-L)
Pn(c|ab) = 1-(1-cb)(1-L)
Pn(c|ab) = 1-(1-ca)(1-L)
Pn(c|ab) = 1-(1-L)
= L = 0.3
Pn(c|b)=Pn(c|ab)P(a)+Pn(c|ab)P(a)
1-0.7=(1-ca)(1-cb)(1-L)0.1+(1-cb)(1-L)0.9
0.3=0.5(1-cb)0.07+(1-cb)0.7*0.9
=0.035(1-cb)+0.63(1-cb)=0.665(1-cb)
0.55=cb

28. Graph Models Bipartite graphs E.g. medical reasoning
Generally, diseases cause symptom (not reverse) s1 s2 d1 s3 d2 s4 d3 s5 d4 s6

29. Topologies Generally more complex General Bayes Nets
Polytree: One path between any two nodes
General Bayes Nets
Graphs with undirected cycles
No directed cycles - can’t be own cause
Issue: Automatic net acquisition
Update probabilities by observing data
Learn topology: use statistical evidence of indep, heuristic search to find most probable structure

30. Holmes Example (Pearl)
Holmes is worried that his house will be burgled. For
the time period of interest, there is a 10^-4 a priori chance
of this happening, and Holmes has installed a burglar alarm
to try to forestall this event. The alarm is 95% reliable in
sounding when a burglary happens, but also has a false
positive rate of 1%. Holmes’ neighbor, Watson, is 90% sure
to call Holmes at his office if the alarm sounds, but he is also
a bit of a practical joker and, knowing Holmes’ concern,
might (30%) call even if the alarm is silent. Holmes’ other
neighbor Mrs. Gibbons is a well-known lush and often
befuddled, but Holmes believes that she is four times more
likely to call him if there is an alarm than not.

31. Holmes Example: Model There a four binary random variables:
B: whether Holmes’ house has been burgled
A: whether his alarm sounded
W: whether Watson called
G: whether Gibbons called W B A G

32. Holmes Example: Tables
B = #t B=#f A #t #f
W=#t W=#f
A=#t A=#f B #t #f A #t #f
G=#t G=#f

33. Decision Making Design model of rational decision making
Maximize expected value among alternatives
Uncertainty from
Outcomes of actions
Choices taken
To maximize outcome
Select maximum over choices
Weighted average value of chance outcomes

34. Gangrene Example Medicine Amputate foot Worse 0.25 Full Recovery 0.7
1000
Die 0.05
Die 0.01
Live 0.99
850
Medicine
Amputate leg
Live 0.6
995
Live 0.98
700
Die 0.4
Die 0.02

35. Decision Tree Issues Problem 1: Tree size Solution 1: Hill-climbing
k activities : 2^k orders
Solution 1: Hill-climbing
Choose best apparent choice after one step
Use entropy reduction
Problem 2: Utility values
Difficult to estimate, Sensitivity, Duration
Change value depending on phrasing of question
Solution 2c: Model effect of outcome over lifetime

36. Conclusion Reasoning with uncertainty Bayes’ Nets Decision Trees
Many real systems uncertain - e.g. medical diagnosis
Bayes’ Nets
Model (in)dependence relations in reasoning
Noisy-OR simplifies model/computation
Assumes causes independent
Decision Trees
Model rational decision making
Maximize outcome: Max choice, average outcomes

37. Holmes Example (Pearl)
Holmes is worried that his house will be burgled. For
the time period of interest, there is a 10^-4 a priori chance
of this happening, and Holmes has installed a burglar alarm
to try to forestall this event. The alarm is 95% reliable in
sounding when a burglary happens, but also has a false
positive rate of 1%. Holmes’ neighbor, Watson, is 90% sure
to call Holmes at his office if the alarm sounds, but he is also
a bit of a practical joker and, knowing Holmes’ concern,
might (30%) call even if the alarm is silent. Holmes’ other
neighbor Mrs. Gibbons is a well-known lush and often
befuddled, but Holmes believes that she is four times more
likely to call him if there is an alarm than not.

38. Holmes Example: Model There a four binary random variables:
B: whether Holmes’ house has been burgled
A: whether his alarm sounded
W: whether Watson called
G: whether Gibbons called W B A G

39. Holmes Example: Tables
B = #t B=#f A #t #f
W=#t W=#f
A=#t A=#f B #t #f A #t #f
G=#t G=#f